"We don't know who invented the recorder; we do know that for centuries it has been also been known as the English flute"

I don't think we know any such thing. Here's what some bloke on that Internet says:

The true recorder only came into extensive use around 1500. With its versatility and wide, chromatic range, it quickly established itself as the "polite" whistle, for use by professionals and courtiers, whereas the old type of whistle was regarded as a country instrument fit only for shepherds. Henry VIII had a particular affection for recorders ... By Elizabeth I's reign, fashion had changed, and the flute was more popular. The new operas, which arose around 1600, used both. The recorder was particularly associated with pastoral subjects - nymphs, shepherds and so on. The flute was the more "mainstream"orchestral instrument. This continued until the mid to late seventeenth century, when the recorder once more came into vogue as a concert instrument in England - so much so that the "English Flute", or just "Flute" in England, was the recorder, and the cross-blown version was known as the "German Flute". During the first half of the eighteenth century, the German flute grew in popularity, and gradually displaced the English Flute. By 1800 the recorder had virtually disappeared from the professional scene.

So that's

1500ish: recorder appears
1600ish: recorder unfashionable
Mid- to late 1600s: recorder back in fashion; known as 'English Flute'
Mid- to late 1700s: recorder unfashionable again
1800ish: recorder disappears